Evolution of matter fields in black holes with Lifshitz symmetry

Abstract

In this work the evolution of two fields of matter in planar symmetric black holes D-dimensional with symmetry of Lifshitz whose dynamic exponent is $z$ were analyzed. The fields investigated were a scalar field non-minimal coupled to the Einstein tensor and the Ricci scalar and the electromagnetic field. Two black holes were chosen, one with 5-dimensions and z=1 and another with 6-dimensions and $z=0$. The equations of motion for both fields were developed in general and applied to each black hole mentioned. In some cases exact solutions of the equations were obtained in terms of hypergeometric functions and confluent Heun functions. The quasinormal modes (MQN) of evolution of the analyzed fields were calculated numerically using two different approaches: HH and AIM. In all the black holes studied the NSM's did not indicate instabilities neither regarding the couplings nor the space-times. In general, the quasinormal modes behave like the oscillation modes of a damped harmonic oscillator, presenting three regimes: underdamped $(\omega_R \neq0, \omega_I <0) $, critically damped $ (\omega_R=0,\omega_I^{crit} <0) $ and overdamped $ (\omega_R=0,\omega_I <0) $. In the analyzed cases an interesting behavior was found for $\omega_I $ when $ k \neq0 $ and $ r_+<10 $. In general, $\omega_I $ increase with $k$, but here we find a quadratic growth for $ k\sim r_+$ and then a decrease when $k>>r_+$. This behavior may be relevant in the context of gravity/gauge duality. Additionally, we also analyzed the evolution of the same fields in two black holes with symmetry of Lifshtiz in $ 2 + 1 $ dimensions.